Efficient multivariate approximation with transformed rank-1 lattices

We study the approximation of functions defined on different domains by trigonometric and transformed trigonometric functions. We investigate which of the many results known from the approximation theory on the d-dimensional torus can be transfered to other domains. We define invertible parameterized transformations and prove conditions under which functions from a weighted Sobolev space can be transformed into functions defined on the torus, that still have a certain degree of Sobolev smoothness and for which we know worst-case upper error bounds. By reverting the initial change of variables we transfer the fast algorithms based on rank-1 lattices used to approximate functions on the torus efficiently over to other domains and obtain adapted FFT algorithms.:1 Introduction 2 Preliminaries and notations 3 Fourier approximation on the torus 4 Torus-to-R d transformation mappings 5 Torus-to-cube transformation mappings 6 Conclusion Alphabetical IndexWir betrachten die Approximation von Funktionen, die auf verschiedenen Gebieten definiert sind, mittels trigonometrischer und transformierter trigonometrischer Funktionen. Wir untersuchen, welche bisherigen Ergebnisse für die Approximation von Funktionen, die auf einem d-dimensionalen Torus definiert wurden, auf andere Definitionsgebiete übertragen werden können. Dazu definieren wir parametrisierte Transformationsabbildungen und beweisen Bedingungen, bei denen Funktionen aus einem gewichteten Sobolevraum in Funktionen, die auf dem Torus definiert sind, transformiert werden können, die dabei einen gewissen Grad an Sobolevglattheit behalten und für die obere Schranken der Approximationsfehler bewiesen wurden. Durch Umkehrung der ursprünglichen Koordinatentransformation übertragen wir die schnellen Algorithmen, die Rang-1 Gitter Methoden verwenden um Funktionen auf dem Torus effizient zu approximieren, auf andere Definitionsgebiete und erhalten adaptierte FFT Algorithmen.:1 Introduction 2 Preliminaries and notations 3 Fourier approximation on the torus 4 Torus-to-R d transformation mappings 5 Torus-to-cube transformation mappings 6 Conclusion Alphabetical Inde

Fast Fourier Transform at Nonequispaced Nodes and Applications

The direct computation of the discrete Fourier transform at arbitrary nodes requires O(NM) arithmetical operations, too much for practical purposes. For equally spaced nodes the computation can be done by the well known fast Fourier transform (FFT) in only O(N log N) arithmetical operations. Recently, the fast Fourier transform for nonequispaced nodes (NFFT) was developed for the fast approximative computation of the above sums in only O(N log N + M log 1/e), where e denotes the required accuracy. The principal topics of this thesis are generalizations and applications of the NFFT. This includes the following subjects: - Algorithms for the fast approximative computation of the discrete cosine and sine transform at nonequispaced nodes are developed by applying fast trigonometric transforms instead of FFTs. - An algorithm for the fast Fourier transform on hyperbolic cross points with nonequispaced spatial nodes in 2 and 3 dimensions based on the NFFT and an appropriate partitioning of the hyperbolic cross is proposed. - A unified linear algebraic approach to recent methods for the fast computation of matrix-vector-products with special dense matrices, namely the fast multipole method, fast mosaic-skeleton approximation and H-matrix arithmetic, is given. Moreover, the NFFT-based summation algorithm by Potts and Steidl is further developed and simplified by using algebraic polynomials instead of trigonometric polynomials and the error estimates are improved. - A new algorithm for the characterization of engineering surface topographies with line singularities is proposed. It is based on hard thresholding complex ridgelet coefficients combined with total variation minimization. The discrete ridgelet transform is designed by first using a discrete Radon transform based on the NFFT and then applying a dual-tree complex wavelet transform. - A new robust local scattered data approximation method is introduced. It is an advancement of the moving least squares approximation (MLS) and generalizes an approach of van den Boomgard and van de Weijer to scattered data. In particular, the new method is space and data adaptive

Sparse Modelling and Multi-exponential Analysis

The research fields of harmonic analysis, approximation theory and computer algebra are seemingly different domains and are studied by seemingly separated research communities. However, all of these are connected to each other in many ways. The connection between harmonic analysis and approximation theory is not accidental: several constructions among which wavelets and Fourier series, provide major insights into central problems in approximation theory. And the intimate connection between approximation theory and computer algebra exists even longer: polynomial interpolation is a long-studied and important problem in both symbolic and numeric computing, in the former to counter expression swell and in the latter to construct a simple data model. A common underlying problem statement in many applications is that of determining the number of components, and for each component the value of the frequency, damping factor, amplitude and phase in a multi-exponential model. It occurs, for instance, in magnetic resonance and infrared spectroscopy, vibration analysis, seismic data analysis, electronic odour recognition, keystroke recognition, nuclear science, music signal processing, transient detection, motor fault diagnosis, electrophysiology, drug clearance monitoring and glucose tolerance testing, to name just a few. The general technique of multi-exponential modeling is closely related to what is commonly known as the Padé-Laplace method in approximation theory, and the technique of sparse interpolation in the field of computer algebra. The problem statement is also solved using a stochastic perturbation method in harmonic analysis. The problem of multi-exponential modeling is an inverse problem and therefore may be severely ill-posed, depending on the relative location of the frequencies and phases. Besides the reliability of the estimated parameters, the sparsity of the multi-exponential representation has become important. A representation is called sparse if it is a combination of only a few elements instead of all available generating elements. In sparse interpolation, the aim is to determine all the parameters from only a small amount of data samples, and with a complexity proportional to the number of terms in the representation. Despite the close connections between these fields, there is a clear lack of communication in the scientific literature. The aim of this seminar is to bring researchers together from the three mentioned fields, with scientists from the varied application domains.Output Type: Meeting Repor